Spectral dimensions of Krein--Feller operators and $L^{q}$-spectra

TL;DR

This paper investigates the spectral dimensions of Krein-Feller operators linked to arbitrary measures, establishing connections with multifractal spectra and providing conditions for their existence, with applications to various measure types.

Contribution

It introduces new relationships between spectral dimensions, $L^{q}$-spectra, and multifractal measures, and offers conditions for spectral dimension existence, including explicit examples.

Findings

Upper spectral dimension equals the fixed point of the $L^{q}$-spectrum.

Bounds relate spectral and Minkowski dimensions.

Confirmed spectral dimension existence for self-conformal and pure point measures.

Abstract

We study the spectral dimensions and spectral asymptotics of Krein-Feller operators for arbitrary finite Borel measures on $\left(0,1\right).$ Connections between the spectral dimension, the $L^{q}$-spectrum, the partition entropy and the optimised coarse multifractal dimension are established. In particular, we show that the upper spectral dimension always corresponds to the fixed point of the $L^{q}$-spectrum of the corresponding measure. Natural bounds reveal intrinsic connections to the Minkowski dimension of the support of the associated Borel measure. Further, we give a sufficient condition on the $L^{q}$-spectrum to guarantee the existence of the spectral dimension. As an application, we confirm the existence of the spectral dimension of self-conformal measures with or without overlap as well as of certain measures of pure point type. We construct a simple example for which the spectral dimension does not exist and determine explicitly its upper and lower spectral dimension.